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In mathematics, the Marcinkiewicz interpolation theorem, discovered by , is a result bounding the norms of non-linear operators acting on ''L''p spaces. Marcinkiewicz' theorem is similar to the Riesz–Thorin theorem about linear operators, but also applies to non-linear operators. ==Preliminaries== Let ''f'' be a measurable function with real or complex values, defined on a measure space (''X'', ''F'', ω). The distribution function of ''f'' is defined by : Then ''f'' is called weak if there exists a constant ''C'' such that the distribution of ''f'' satisfies the following inequality for all ''t'' > 0: : The smallest constant ''C'' in the inequality above is called the weak norm and is usually denoted by ||''f''||1,''w'' or ||''f''||1,∞. Similarly the space is usually denoted by ''L''1,''w'' or ''L''1,∞. (Note: This terminology is a bit misleading since the weak norm does not satisfy the triangle inequality as one can see by considering the sum of the functions on given by and , which has norm 4 not 2.) Any function belongs to ''L''1,''w'' and in addition one has the inequality : This is nothing but Markov's inequality (aka Chebyshev's Inequality). The converse is not true. For example, the function 1/''x'' belongs to ''L''1,''w'' but not to ''L''1. Similarly, one may define the weak space as the space of all functions ''f'' such that belong to ''L''1,''w'', and the weak norm using : More directly, the ''L''''p'',''w'' norm is defined as the best constant ''C'' in the inequality : for all ''t'' > 0. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Marcinkiewicz interpolation theorem」の詳細全文を読む スポンサード リンク
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